There's no need for ac in the basement. However, it will be needed for the first floor, so you'll end up with a forced air system there in addition to the radiant. Costs are local, so check on relative installation expenses.

We had radiant in-floor heat on both levels of our home - main level was on a slab and they poured a thin layer of Gypcrete over the tubes on the 2nd floor - I wouldn’t have had it any other way! (Our primary BR was on the 2nd floor). Put a separate thermostat in the bathrooms (from the bedrooms - as folks typically want the bedrooms to be a bit cooler when they sleep). This was a mountain home in CO, so there was no need for A/C.

If you are using a forced air heating and cooling system on the main floor of the house, radiant heat in the basement works great because it’s too hard to regulate the forced air heat down there. (Usually the supplies are in the ceiling and to get the heat to go down to the floor is contrary to thermal dynamics). Forced air systems are usually very cost effective and gives the option for air conditioning, so I would use that for the majority of the house and use radiant in floor hot water heat with an electric boiler for the basement. (It will also give you a drier basement, especially if you open the A/C vents in the summer).

Julie - There really isn't that much temperature stratification to worry about. I have ducts in floors and ceilings of various houses and I don't really care where they are in heating or a/c season.

And explain to me how radiant heat makes for a drier basement? Especially if you are talking about opening a/c vents? How does that correlate to radiant heat?

If radiant were to leak, that would be better under basement slab - I suppose that is one reason. The diminished need for a/c in a basement would be another. Concrete slab is the cheapest way to do radiant is another reason.

If you insulate the basement well (and you should), then the heat load is pretty low (few windows, conditioned space above, minimal air leakage). So the radiant is limping along and seems like a bit of overkill.

Radiant is expensive. Better to spend that money to make the structure tight and barely need heat. More comfortable too. If you have endless money, then do both. But the best emphasis should be on a tight structure.

David, I’m not sure you understand me and I’m not sure I understand you. Radiant in floor heat in the basement will give you a warmer and drier basement because the heat is at your feet, moving upward and the warm slab is drying out one of the dampest surfaces in the basement. If the basement is used only for storage and tornados, then it’s an expensive upgrade, but if you plan on spending any time in the winter down there, it’s well worth it.
I would hope in the summer the radiant system would be turned off and the forced air A/C ducts opened to allow some dehumidification of that space, there again, the dampest part of a house in the summer months.
The question was if they should put radiant heat in just the basement or the basement and the main floor. I vote basement only because of costs. I also believe every new house should have a blower door test with a result about 2.5 ACH @ 50 pa.